How can I change between radian and degree mode?
Press the menu button in the upper right corner (three dots). Tick the “Use radians” checkbox if you want to have radians. If unchecked Algeo will use degrees. Note however that graphing is always in radian mode.

How can I make the cursor jump out from under the square root?
Tap on a whitespace right to the square root sign. The cursor jumps to the normal level. This works as well with powers. If you don’t like this behaviour you can change the input style to the old mode. This won’t typeset the fractions, powers and roots so the cursor won’t stuck inside a square root. To enable this mode go to Menu->Settings and check ‘Simple input method’.

Where’s inverse sin and cos?
They’re called arcsin and arccos. The other trigonometric functions have similar names.

How can I calculate arbitrary based logarithms?
You can calculate natural and 10-based logarithm with ln and log. To calculate the a based logarithm of 3 you can do the following:
You can read more about this on wikipedia.